| import OAI.Probability.InvariantIsing.Pressure.PhysicalPressureMean |
| import OAI.Probability.InvariantIsing.Pressure.PhysicalPressureConcentration |
|
|
| namespace OAI |
|
|
| /-! Manuscript Theorem 1.1, conditional only on the explicitly cited |
| published concentration and finite hierarchical field inputs. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology |
|
|
| namespace InvariantIsing |
|
|
| theorem limiting_pressure |
| (hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput) |
| (hpub : PanchenkoTalagrandFieldPairInput) |
| {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] |
| (U : (N : ℕ) → Ω → Orthogonal N) (hU : ∀ N, Measurable (U N)) |
| (hHaar : ∀ N, (P.map (U N)).IsMulRightInvariant) |
| (eig : (N : ℕ) → Fin N → ℝ) (ν : ProbabilityMeasure ℝ) (a b : ℝ) |
| (hcompact : IsCompact (ν : Measure ℝ).support) |
| (hbound : (ν : Measure ℝ).support ⊆ Icc a b) |
| (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support) |
| (hno : ∀ ε : ℝ, 0 < ε → ∀ᶠ N in atTop, ∀ i, a-ε ≤ eig N i ∧ eig N i ≤ b+ε) |
| (hweak : Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1))) |
| atTop (𝓝 ν)) : |
| Tendsto (fun N => ∫ ω, rotatedPressure (eig N) (matrixRotation (U N ω)⁻¹) (fun _ => 0) ∂P) |
| atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) ∧ |
| ∀ᵐ ω ∂P, Tendsto (fun N => rotatedPressure (eig N) (matrixRotation (U N ω)⁻¹) (fun _ => 0)) |
| atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) := by |
| refine ⟨physical_mean_pressure_tendsto hhaar hgauss hpub P U hU hHaar eig ν a b |
| hcompact hbound ha hb hno hweak,?_⟩ |
| have hab : a ≤ b := (hbound ha).2 |
| let clip := fun N (i : Fin N) => spectralClip a b (eig N i) |
| have hcmean := physical_mean_pressure_tendsto hhaar hgauss hpub P U hU hHaar clip ν a b |
| hcompact hbound ha hb (fun ε hε => Eventually.of_forall (fun N i => by |
| have hh := spectralClip_mem hab (eig N i) |
| constructor <;> linarith [hh.1,hh.2])) |
| (spectral_clip_empirical_tendsto eig ν a b hbound hweak) |
| let K := |a|+|b|+1 |
| have hK : 0 < K := by dsimp [K]; positivity |
| have hclip N i : |clip N i| ≤ K := by |
| have hh := spectralClip_mem hab (eig N i) |
| rw [abs_le] |
| dsimp [K] |
| constructor <;> linarith [hh.1,hh.2,neg_abs_le a,le_abs_self b,abs_nonneg a,abs_nonneg b] |
| have hconc := ae_physical_pressure_sub_mean_tendsto_zero hhaar P |
| (fun n => U (n+3)) (fun n => hU (n+3)) (fun n => hHaar (n+3)) |
| (fun n => clip (n+3)) (fun _ _ => 0) K hK (fun n => hclip (n+3)) |
| have hη := (spectralExcess_tendsto eig a b hno).comp (tendsto_add_atTop_nat 3) |
| filter_upwards [hconc] with ω hω |
| have hclim := hω.add (hcmean.comp (tendsto_add_atTop_nat 3)) |
| have hd : Tendsto (fun n => rotatedPressure (eig (n+3)) (matrixRotation (U (n+3) ω)⁻¹) (fun _ => 0)- |
| rotatedPressure (clip (n+3)) (matrixRotation (U (n+3) ω)⁻¹) (fun _ => 0)) |
| atTop (𝓝 0) := by |
| apply squeeze_zero_norm _ (by simpa only [zero_div] using hη.div_const 2) |
| intro n |
| rw [Real.norm_eq_abs,abs_sub_comm] |
| exact abs_rotatedPressure_sub_le (by omega) (clip (n+3)) (eig (n+3)) |
| (matrixRotation (U (n+3) ω)⁻¹) (fun _ => 0) (spectralExcess (eig (n+3)) a b) |
| (fun i => spectralClip_close hab (spectralExcess_nonneg _ _ _) (spectralExcess_bounds _ _ _ i)) |
| apply (tendsto_add_atTop_iff_nat 3).mp |
| have hh := hclim.add hd |
| convert hh using 1 |
| · funext n |
| dsimp only [Function.comp_apply] |
| ring |
| · simp only [zero_add,add_zero] |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
